2
votes
1answer
52 views

Distance between two points in the plane

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance ...
0
votes
3answers
90 views

If a triangle has rational coordinates, does it have rational area?

Basically, the topic says it all. If a triangle has rational coordinates (say, in $\mathbb Q^2$), must it have rational area? I realize the side-lengths are usually irrational; that's fine. Heron's ...
0
votes
1answer
135 views

Drawing a triangle in a unit circle

This is a question that I derived for a long time ago. It asks if we draw a triangle in a unit circle does all arc lengths $(\alpha ,\beta ,\theta)$ and sides of triangle $(a,b,c)$ can be rational ...
18
votes
1answer
382 views

How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?

Open problem in Geometry/Number Theory. The real question here is: Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational? The ...
7
votes
2answers
226 views

Chances of avoiding the diagonal

A circle of radius 1 is randomly placed in a rectangle $ABCD$ so that the circle lies completely inside the rectangle. Length and breadth of rectangles are 36 and 15 respectively. Let the ...